In a band-grinding machine of this type, which includes a motor-operable contact disk, it is known from German Pat. No. DE 2400267 to make use of a rerouting roller, which has a relatively low mass compared to its diameter, and to provide a band-tensioning device, which at the operating velocity of the grinding band, subjects the grinding band to a low force of about 0.5 to 5N. The rerouting roller is formed thereat as a spoke wheel, the periphery of which is formed as a smooth circular running surface. So as to avoid any jumping off of the grinding band at high velocities of travel, the running surface of the rerouting roller, as well as that of the contact disk is outwardly convexly curved in a direction at right angles to the longitudinal direction of the band. But in order to prevent the grinding band from jumping off also during starting, the grinding band is initially adjusted to an increased tension with the aid of a band tensioning device and/or by means of an adjustment of the axial spacing between the rerouting roller and the contact disk. After the required band velocity has been reached, which is suitable for grinding, the band tension is reduced thereafter to a somewhat lower value suitable for normal operation. When the band is gradually or abruptly brought to a halt, it is hard, however, to avoid any further rotation of the rerouting roller, in spite of its low weight, and to prevent the grinding band from coming off therefrom.